Дано: АМ=ЕК. Доказать:∆АКО=∆ЕМО

Proof:

Given: AM = EK

To Prove: ΔAKO = ΔEMO

Proof:

We will prove that ΔAKO = ΔEMO using the Side-Side-Side (SSS) congruence criterion.

We know that:

1. AM = EK (Given)
2. AO = EO (By definition of midepoint)
3. ∠MAK = ∠MEK (By definition of midepoint)

Now, we need to show that OK = MO to apply the SSS congruence criterion.

From 2, we have:

AO = EO

Adding MO to both sides:

AO + MO = EO + MO

Since AM = EK (Given), we can substitute AM for EK:

AO + MO = AM + MO

And since AO = EO (By definition of midepoint):

EO + MO = AM + MO

Subtracting MO from both sides:

EO = AM

So now, we have EO = AM, and from 3, we have ∠MAK = ∠MEK.

Therefore, we can apply the SSS congruence criterion:

ΔAKO = ΔEMO (By SSS criterion)

Thus, we have proven that ΔAKO is congruent to ΔEMO.